Square pyramidal number
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In mathematics, a pyramid number, or square pyramidal number, is a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
that counts the number of stacked spheres in a
pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
with a square base. The study of these numbers goes back to
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
and
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...
. They are part of a broader topic of
figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polyg ...
s representing the numbers of points forming regular patterns within different shapes. As well as counting spheres in a pyramid, these numbers can be described algebraically as a sum of the first n positive
square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s, or as the values of a
cubic polynomial In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
. They can be used to solve several other counting problems, including counting squares in a square grid and counting
acute triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...
s formed from the vertices of an odd
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
. They equal the sums of consecutive
tetrahedral number A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is, ...
s, and are one-fourth of a larger tetrahedral number. The sum of two consecutive square pyramidal numbers is an
octahedral number In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The ''n''th octahedral number O_n can be obtained by the formula:. :O_n=. The first few octahed ...
.


History

The pyramidal numbers were one of the few types of three-dimensional figurate numbers studied in
Greek mathematics Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
, in works by
Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...
,
Theon of Smyrna Theon of Smyrna ( el, Θέων ὁ Σμυρναῖος ''Theon ho Smyrnaios'', ''gen.'' Θέωνος ''Theonos''; fl. 100 CE) was a Greek philosopher and mathematician, whose works were strongly influenced by the Pythagorean school of thought. Hi ...
, and
Iamblichus Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
. Formulas for summing consecutive squares to give a cubic polynomial, whose values are the square pyramidal numbers, are given by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
, who used this sum as a lemma as part of a study of the volume of a
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
, and by
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...
, as part of a more general solution to the problem of finding formulas for sums of progressions of squares. The square pyramidal numbers were also one of the families of figurate numbers studied by
Japanese mathematicians Japanese may refer to: * Something from or related to Japan, an island country in East Asia * Japanese language, spoken mainly in Japan * Japanese people, the ethnic group that identifies with Japan through ancestry or culture ** Japanese diaspo ...
of the wasan period, who named them "kirei saijō suida" (with modern
kanji are the logographic Chinese characters taken from the Chinese family of scripts, Chinese script and used in the writing of Japanese language, Japanese. They were made a major part of the Japanese writing system during the time of Old Japanese ...
, 奇零 再乗 蓑深). The same problem, formulated as one of counting the
cannonball A round shot (also called solid shot or simply ball) is a solid spherical projectile without explosive charge, launched from a gun. Its diameter is slightly less than the bore of the barrel from which it is shot. A round shot fired from a lar ...
s in a square pyramid, was posed by
Walter Raleigh Sir Walter Raleigh (; – 29 October 1618) was an English statesman, soldier, writer and explorer. One of the most notable figures of the Elizabethan era, he played a leading part in English colonisation of North America, suppressed rebelli ...
to mathematician
Thomas Harriot Thomas Harriot (; – 2 July 1621), also spelled Harriott, Hariot or Heriot, was an English astronomer, mathematician, ethnographer and translator to whom the theory of refraction is attributed. Thomas Harriot was also recognized for his cont ...
in the late 1500s, while both were on a sea voyage. The
cannonball problem In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be a ...
, asking whether there are any square pyramidal numbers that are also square numbers other than 1 and 4900, is said to have developed out of this exchange.
Édouard Lucas __NOTOC__ François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him. Biography Lucas ...
found the 4900-ball pyramid with a square number of balls, and in making the cannonball problem more widely known, suggested that it was the only nontrivial solution. After incomplete proofs by Lucas and Claude-Séraphin Moret-Blanc, the first complete proof that no other such numbers exist was given by G. N. Watson in 1918.


Formula

If spheres are packed into square pyramids whose number of layers is 1, 2, 3, etc., then the square pyramidal numbers giving the numbers of spheres in each pyramid are: These numbers can be calculated algebraically, as follows. If a pyramid of spheres is decomposed into its square layers with a square number of spheres in each, then the total number P_n of spheres can be counted as the sum of the number of spheres in each square, P_n = \sum_^nk^2 = 1 + 4 + 9 + \cdots + n^2, and this
summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...
can be solved to give a
cubic polynomial In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
, which can be written in several equivalent ways: P_n= \frac = \frac = \frac + \frac + \frac. This equation for a sum of squares is a special case of
Faulhaber's formula In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the ''p''-th powers of the first ''n'' positive integers :\sum_^n k^p = 1^p + 2^p + 3^p + \cdots + n^p as a (''p''&nb ...
for sums of powers, and may be proved by
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
. More generally,
figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polyg ...
s count the numbers of geometric points arranged in regular patterns within certain shapes. The centers of the spheres in a pyramid of spheres form one of these patterns, but for many other types of figurate numbers it does not make sense to think of the points as being centers of spheres. In modern mathematics, related problems of counting points in integer polyhedra are formalized by the
Ehrhart polynomial In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a highe ...
s. These differ from figurate numbers in that, for Ehrhart polynomials, the points are always arranged in an
integer lattice In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid ...
rather than having an arrangement that is more carefully fitted to the shape in question, and the shape they fit into is a polyhedron with lattice points as its vertices. Specifically, the Ehrhart polynomial of an integer polyhedron is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
that counts the number of integer points in a copy of that is expanded by multiplying all its coordinates by the number . The usual symmetric form of a square pyramid, with a
unit square In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordin ...
as its base, is not an integer polyhedron, because the topmost point of the pyramid, its apex, is not an integer point. Instead, the Ehrhart polynomial can be applied to an asymmetric square pyramid with a unit square base and an apex that can be any integer point one unit above the base plane. For this choice of , the Ehrhart polynomial of a pyramid is .


Geometric enumeration

As well as counting spheres in a pyramid, these numbers can be used to solve several other counting problems. For example, a common
mathematical puzzle Mathematical puzzles make up an integral part of recreational mathematics. They have specific rules, but they do not usually involve competition between two or more players. Instead, to solve such a puzzle, the solver must find a solution that sati ...
involves finding the number of squares in a large by square grid. This number can be derived as follows: *The number of squares found in the grid is . *The number of squares found in the grid is . These can be counted by counting all of the possible upper-left corners of squares. *The number of squares found in the grid is . These can be counted by counting all of the possible upper-left corners of squares. It follows that the number of squares in an square grid is: n^2 + (n-1)^2 + (n-2)^2 + (n-3)^2 + \ldots = \frac. That is, the solution to the puzzle is given by the th square pyramidal number. The number of rectangles in a square grid is given by the
squared triangular number In number theory, the sum of the first cubes is the square of the th triangular number. That is, :1^3+2^3+3^3+\cdots+n^3 = \left(1+2+3+\cdots+n\right)^2. The same equation may be written more compactly using the mathematical notation for summa ...
s. The square pyramidal number P_n also counts the number of
acute triangle An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...
s formed from the vertices of a (2n+1)-sided
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
. For instance, an equilateral triangle contains only one acute triangle (itself), a regular
pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
has five acute
golden triangle Golden Triangle may refer to: Places Asia * Golden Triangle (Southeast Asia), named for its opium production * Golden Triangle (Yangtze), China, named for its rapid economic development * Golden Triangle (India), comprising the popular tourist ...
s within it, a regular
heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than ''hepta-'', a Greek-derived num ...
has 14 acute triangles of two shapes, etc. More abstractly, when permutations of the rows or columns of a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
are considered as equivalent, the number of 2\times 2 matrices with non-negative integer coefficients summing to n, for odd values of n, is a square pyramidal number.


Relations to other figurate numbers

The
cannonball problem In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be a ...
asks for the sizes of pyramids of cannonballs that can also be spread out to form a square array, or equivalently, which numbers are both square and square pyramidal. Besides 1, there is only one other number that has this property: 4900, which is both the 70th square number and the 24th square pyramidal number. The square pyramidal numbers can be expressed as sums of
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s: P_n = \binom + \binom. The binomial coefficients occurring in this representation are
tetrahedral number A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is, ...
s, and this formula expresses a square pyramidal number as the sum of two tetrahedral numbers in the same way as square numbers are the sums of two consecutive
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
s. If a tetrahedron is reflected across one of its faces, the two copies form a
triangular bipyramid In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, i ...
. The square pyramidal numbers are also the figurate numbers of the triangular bipyramids, and this formula can be interpreted as an equality between the square pyramidal numbers and the triangular bipyramidal numbers. Analogously, reflecting a square pyramid across its base produces an octahedron, from which it follows that each
octahedral number In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The ''n''th octahedral number O_n can be obtained by the formula:. :O_n=. The first few octahed ...
is the sum of two consecutive square pyramidal numbers. Square pyramidal numbers are also related to tetrahedral numbers in a different way: the points from four copies of the same square pyramid can be rearranged to form a single tetrahedron with twice as many points along each edge. That is, 4P_n=Te_=\binom. To see this, arrange each square pyramid so that each layer is directly above the previous layer, e.g. the heights are
4321
3321
2221
1111
Four of these can then be joined by the height pillar to make an even square pyramid, with layers 4, 16, 36, \dots. Each layer is the sum of consecutive triangular numbers, i.e. (1+3), (6+10), (15+21), \dots, which, when totalled, sum to the tetrahedral number.


Other properties

The
alternating series In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternatin ...
of
unit fraction A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc ...
s with the square pyramidal numbers as denominators is closely related to the Leibniz formula for , although it converges more quickly. It is: \begin \sum_^& (-1)^\frac\\ &=1-\frac+\frac-\frac+\frac-\frac+\frac-\frac+\cdots\\ &=6(\pi-3)\\ &\approx 0.849556.\\ \end In
approximation theory In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...
, the sequences of odd numbers, sums of odd numbers (square numbers), sums of square numbers (square pyramidal numbers), etc., form the coefficients in a method for converting
Chebyshev approximation In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' wi ...
s into
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s.


References


External links

* {{Classes of natural numbers Figurate numbers Pyramids Articles containing video clips